Rowing has been at Oakwell for the past nine years, starting as general manager before becoming a director and then chief executive.
"I have thoroughly enjoyed my time with Barnsley and will miss working with the staff and my interaction with the fans," he told the club's official website.
"I have particularly enjoyed working with Patrick Cryne (Tykes owner), but as his involvement with the club has gradually diminished and with my recent 70th birthday, I think it's time to blow the final whistle.
"I will continue to support the club as a season ticket holder and wish the club and its fans every success."
Cryne added: "Don has been the unsung hero at the club. In his position you draw fire for all the poor decisions made within the club, but get little credit for the good work done.
"His workload has always been phenomenal and yet he has discharged it completely. I regard him as a good friend and wish him a long and fruitful retirement."